When it comes to fasting the six days of Shawwal when someone has days to make up from Ramadan, then the Scholars differed if one can precede the recommended fasts over the obligatory fasts. However, the strongest opinion is that one should make up the obligatory fasts from Ramadan before beginning to fast the six days of Shawwal due to the fact the Hadith states:

‎مَنْ صَامَ رَمَضَانَ, ثُمَّ أَتْبَعَهُ سِتًّا مِنْ شَوَّالٍ كَانَ كَصِيَامِ اَلدَّهْرِ

Whoever fasts during the month of Ramadan and then follows it with six days of Shawwal will be (rewarded) as if he had fasted the entire year [Muslim]

So a person needs to first finish Ramadan and then follow it with six days of Shawwal in order to obtain the reward mentioned above.

Shaikh Bin Baz (may Allah have mercy upon him) was asked:

Question:

“Is it permissible to observe Sawm (Fasting) for six days in Shawwal before making up for the missed days of the previous month of Ramadan?”

Answer:

“Scholars differ regarding this issue; however, the soundest opinion is that it is prescribed to give precedence to observing Sawm of the missed days of Ramadan over the six days of Shawwal and other supererogatory Sawm, because the Prophet (peace be upon him) said: Anyone who fasts Ramadan and then follows it with six days of Shawwal it is as if they fasted the whole year. (Related by Muslim in his Sahih [Authentic Hadith Book]) If one chooses to observe Sawm of the six days of Shawwal first, it will not be considered as one who fasts these six days after fasting all of Ramadan, but after some days of Ramadan. Making up for the missed days of Ramadan is a Fard (obligatory act of worship) while fasting the six days of Shawwal is voluntary, and a Fard is more important to fulfill. May Allah grant us success!”. End quote. [Reference]

Shaikh al-Albani (may Allah have mercy upon him) said:

“So we say that the woman who has some days of Ramadan owing on her and then busies herself with the fasting the six days of Shawwal, and delays making up the fasts of Ramadan, then this is a grave mistake. This is because she attempts to draw near to Allah with what He did not obligate upon her and is neglectful of what He has obligated upon her…” End quote. [Silsiltul-Huda wan-Nur (no. 165)]

This is also the opinion of Shaikh Ibn ul-‘Uthaymeen (may Allah have mercy upon him) who said:

“If a person had some fasting to make up for, and he fasted the six days, should he fast them before the days of Ramadan (which he has to make up for) or after them? For example, this man fasted twentyfour days of Ramadan and there remained six days incumbent upon him; so if he fasted the six days of Shawwal before fasting the six days which he had to make up for, then it would not be said that he had fasted Ramadan then followed it by six days from Shawwal. It could not be said that he had fasted Ramadan unless he had fasted all of it. So based upon this, the reward for fasting the six days from Shawwal would not be confirmed for one who fasted them while he still had days to make up for from Ramadan. This question does not belong to the same category as the question of whether it is permissible for one to perform voluntary fasts, who has days of Ramadan to make up for in which there is a difference of opinion among scholars. Because this difference of opinion is not regarding the six days. As for the six days, they follow on from Ramadan and it is not possible for the reward of them to be confirmed except for one who has completed Ramadan” [Fatawa Arkan-ul-Islam (pg. 575-576) of the English translation]

As for the narration of ‘Aisha (may Allah be pleased with her):

‎عَائِشَةَ، – رضى الله عنها – تَقُولُ كَانَ يَكُونُ عَلَىَّ الصَّوْمُ مِنْ رَمَضَانَ فَمَا أَسْتَطِيعُ أَنْ أَقْضِيَهُ إِلاَّ فِي شَعْبَانَ الشُّغُلُ مِنْ رَسُولِ اللَّهِ صلى الله عليه وسلم أَوْ بِرَسُولِ اللَّهِ صلى الله عليه وسلم

‘A’isha (Allah be pleased with her) as saying: I had to complete some of the fasts of Ramadan, but I could not do it but during the month of Sha’ban due to my duties to the Messenger of Allah (ﷺ) or with the Messenger of Allah (ﷺ) [Muslim]

Then there is no evidence here to precede the six days of Shawwal over the fasts of Ramadan.

Shaikh Bin Baz (may Allah have mercy upon him) said:

“As for the questioner’s statement that ‘Aisha (may Allah be pleased with her) delayed the fast until Sha’ban, then she said: ‘Due to my duties to the Messenger of Allah (ﷺ)”. If she delayed the obligatory fasts due to the Messenger of Allah (ﷺ), then it is more deserving and more likely she delayed the recommended fasts due to duties towards the Messenger of Allah (ﷺ). So the conclusion is that there is no evidence in the action of ‘Aisha (may Allah be pleased with her) for preceding the fast of Shawwal over the fast of Ramadan because she delayed the fast of Ramadan due to duties to the Messenger of Allah (ﷺ). So it is more deserving and likely that she delayed the six days of Shawwal. Next, if she did precede the six days of Shawwal [over the missed days of Ramadan], then there is no proof in her action in what opposes the clear texts. Yes”. End quote.  [Reference]

And Allah knows Best

Faisal bin Abdul Qaadir bin Hassan
Abu Sulaymaan